A sonometer wire resonates with a given tuning fork forming standing waves with five antinodes between the two bridges when a mass of $9\,kg$ is suspended from the wire. When this mass is replaced by a mass $M,$ the wire resonates with the same tuning fork forming three antinodes for the same positions of the bridges. The value of $M$ is .... $kg$
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$F = \frac{{\rm{n}}}{{2\ell }}\sqrt {{\rm{T}}/\mu } \quad $ { ${{\rm{n}} \to {{\rm{n}}^{{\rm{th}}}}}$ Harmonic or $n$ atinodes}
In case $(i)$ $\mathrm{f}=\frac{5}{2 \ell} \sqrt{\frac{90}{\mu}}$
In case $(ii)$ $\mathrm{f}=\frac{3}{2 \ell} \sqrt{\frac{\mathrm{T}}{\mu}}$
$\Rightarrow \mathrm{T}=50 \mathrm{\,N}$ or $\mathrm{M}=5 \mathrm{\,kg}$
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